International Journal of Database Theory and Application
Vol.9, No.5 (2016), pp.77-90
http://dx.doi.org/10.14257/ijdta.2016.9.5.08
A Novel Soft Set Approach for Feature Selection
Daoli Yanga, Zhi Xiaoa,b,1,2, Wei Xua, Xianning Wanga and Yuchen Pana
a
School of Economics and Business Administration, Chongqing University,
Chongqing 400044, PR China
b
Chongqing Key Laboratory of Logistics, Chongqing University, Chongqing
400044, PR China
xiaozhicqu@163.com
Abstract
Feature selection is an important preprocess for data mining. The soft set theory is a
new mathematical tool to deal with uncertainties. Merging the soft set theory into the
feature selection process based on rough sets facilitates the computation with equivalence
classes and improves the efficiency. We propose a paired relation soft set model based on
equivalence classes of the information system. Then we use it to present the lower
approximate set in the form of soft sets and calculate the degree of dependency between
relations. Furthermore, we give a new mapping to obtain equivalence classes of
indiscernibility relations and propose a feature selection algorithm based on the paired
relation soft set model. Compared to the algorithm based NSS, this algorithm shows
18.17% improvement on an average. Meantime, both of the algorithms show a good
scalability.
Keywords: soft set, feature selection, rough set, information system
1. Introduction
Feature selection is an important issue in data mining. In the big data era, the scale of
databases grows rapidly with an increasing number of features due to availability of more
detailed information. The existence of irrelative features is inevitable. Feature selection
can eliminate those features so that it reduces the dimensionality, improves the predictive
accuracy and enhances comprehensibility of the induced concepts [1]. Besides, features
may have approximate relationships with each other. Using a subset of features to
represent other features and deduce the classification results is an imprecise process. The
rough set theory is a powerful tool to process such kind of imprecision [2-4]. Thus, it is
incorporated into feature selection algorithms. However, the computation with
equivalence classes in these algorithms is inconvenient and hard to understand. The soft
set theory can facilitate the computation and improve the efficiency.
Recent literatures of rough set based feature selection algorithms mainly focus on three
aspects: feature evaluation, search strategies and practical application. Feature evaluation
provides the significance of the candidate features which is represented by dependency
degree in the rough set theory. For different nature of data, calculation methods of the
dependency degree are different. Several types of data have been studied, such as fuzzy
data [5], data with noise [6], dynamic incomplete data [7], hybrid data [8], etc. As for
search strategies, there are commonly used methods like sequential forward selection and
sequential backward elimination [6], as well as many heuristic methods which are hybrid
genetic algorithm [9], granular neural network [5], water drops algorithm [10], etc. Some
literatures explore feature selection based on rough set for practical application in
1
2
Corresponding Author, email: xiaozhicqu@163.com
Supported by the National Science Foundation of China (Grant No. 71171209)
ISSN: 2005-4270 IJDTA
Copyright ⓒ 2016 SERSC
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
domains like computer security [11] and assessment of power systems [12]. But
researches focusing on the improvement in efficiency when calculating the dependency
degree of features are few. This kind of improvement is also crucial in improving
performance of rough set based feature selection.
The soft set theory was proposed by D. MOLODTSOV in 1999 [13]. It offers a rigid
mathematical theory to deal with uncertainties. Unlike the probability theory and the
fuzzy set theory suffering from the inadequacy of the parameterization tools, the soft set
theory is free of those problems. This makes the theory easily applicable in practice [13]
and very convenient to combine with other theories. In recent years, studies about the soft
set theory are growing rapidly in many fields such as modern algebra [14], forecasting
[15], decision making [16], data analysis [17], etc. It has been proved that every rough set
is a soft set, which shows the necessity to combine two theories together [18]. As for the
dependency degree in the rough set theory, Qin et al. [19] had found an efficient way to
calculate it by constructing a soft set model (NSS) for an information system. Later,
Mamat et al. [20] gave a new clustering method of attributes based on the same soft set
model. Both of them suggest that the method based on the soft set theory is efficient and
easy to apply. However, there are two remaining issues they don’t tackle. One is how to
get equivalence classes of the indiscernibility relation of features based on the soft set
theory and the other is how to apply the soft set model to feature selection which is
different from feature cluster.
Our main contributions are summarized as follows:
(a) We propose a paired relation soft set model (PRSS) based on the information
system. We use it to construct the lower approximate set, present the approximate set in
the form of soft set and furthermore calculate the dependency degree between features.
(b) We propose a method to acquire equivalence classes of indiscernibility relations
based on the PRSS.
(c) We present a feature selection algorithm based on the PRSS and apply it to
seventeen UCI benchmark data sets. The result shows improvement in efficiency
compared to the method based on NSS.
The rest of our paper contains four sections. Section 2 presents the fundamental theory.
Section 3 proposes the paired relation soft set model, a method to acquire equivalence
classes of indiscernibility relations and the relative feature selection algorithm. Section 4
compares experimental results with the NSS method. The final section concludes our
work.
2. Essential Rudiments
2.1. Rough Set Theory
Definition 1 (Information system [4]): Suppose there is a pair =( , ) of non-empty,
finite sets and , where is the universe set of objects, and is a set consisting of
attributes, i.e. functions
, where is the set of values of attribute , called the
domain of . The pair =( , ) is called an information system.
Sometimes, we distinguish in an information system =( , ) a partition of A into
two classes ,
of attributes, called condition and decision (action) attributes,
respectively. The tuple
called a decision system [4].
Definition 2 (Indiscernibility relation [3]): In an information system , if
and
, then
(intersection of all equivalence relations belonging to ), denoted by
, will be called an indiscernibility relation over .
The equivalence class of
involving element
is denoted by
. The
denotes the family of all equivalence classes of
78
Copyright ⓒ 2016 SERSC
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
. Meantime we denote the family of all equivalence relations defined in
as
.
Definition 3 (The lower and upper approximations [3]): With each subset
and
an equivalence relation
, we associate two subsets:
(1)
Called the -lower and -upper approximations of respectively.
Definition 4 (Degree of partial dependency [3]): Let =( , ) be an information
system and ,
. We say that depends in a degree (
) from , if and
only if
(2)
Where
is
denotes the cardinality of the set
and the
-positive region of
.
Dependency degree shows the importance of
for
(3)
.
2.2. Soft Set Theory
Let be the initial universe set and be a set of parameters.
Definition 5 (Soft set [13]): A pair of
is called a soft set (over ) if and only if
is a mapping of into the set of all subset of the set .
In other words, the soft set is a parameterized family of subsets of the set . Every set
,
, from this family may be considered as the set of -elements of the soft set
, or as the set of - approximate elements of the soft set [13].
Definition 6 (Soft subset [21]): For two soft sets
and
over a common
universe set , the
is a soft subset of
if
and
,
and
are identical approximations. It is denoted as
.
Apparently, for a set of - approximate elements of
, we can define a
corresponding soft set
which has only one parameter .
.
Let
and
be soft sets over , then operation * for soft sets [13] is
,
Where
,
,
the sets and .
Example 1: There are five houses
evaluated by three criterions
,
and
, and
(4)
is the Cartesian product of
which are ready to be
. Suppose
. Then we get the soft set
.
If we want to know which houses are both expensive and beautiful, we get
, and execute the intersection operation as follows
(5)
,
(6)
That is
.
2.3. A Soft Set Model on Equivalence Class
Computation in the rough set theory always involves intersection, union of the sets of
equivalence classes and the dependency degree between attributes. It is inconvenient to
execute these operations directly on the information system. Thus, Qin et al. [19]
constructed a soft set model over equivalence classes to facilitate the computation.
Copyright ⓒ 2016 SERSC
79
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
Given an information system =( , ),
denotes the set of all equivalence classes
in the partitions
, where
. Let =
be the initial universe set of objects and
=
be the set of parameters.
denotes the power set of . Meantime, define
mapping
. The pair
is a soft set model over equivalence classes (NSS)
[19]. Table 1 shows the tabular representation of
, where
and
is
either 1 or 0,
,
.
Table 1. The Tabular Representation of the Soft Set over Equivalence
Classes
Then they defined two soft set
and
where
,
:
and
(7)
The lower and upper approximations of parameter
defined as
with respect to attribute
,
where
.
The lower and upper approximations of attribute
defined as
with respect to attribute
are
(8)
are
(9)
where
.
In the following, we just present an example for soft set
.
is similar.
Example 2: we consider the information system of house evaluation as shown in Table
2.
Then, Then they
The lower and upper approximations of parameter with respect to attribute are
defined
h
The lower and upper approximations of attribute with respect to attribute are
he
In the following, we just present an example for soft set
.
is similar.
Example 2: we consider the information system of house evaluation as shown in Table
2.
partitions
of
each
attribution
are
,
and
. The soft set
is shown in Table 3.
For ,
, and ,
,
=
and
= .
Thus,
80
=
.
Copyright ⓒ 2016 SERSC
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
Table 2. An Information System of House Evaluation
price
appearance material
expensive beautiful
wood
cheap
beautiful
metal
cheap
ugly
brick
Table 3. The Tabular Representation of Soft Set
Based on NSS
1
0
1
0
1
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
1
1
0
1
0
1
0
0
0
1
1
0
0
1
0
0
1
0
1
0
0
1
3. Proposed Paired Relation Soft Set Technique
3.1. Paired Relation Soft Set
Given an information system =( , ) and two relations ,
,
.
When examining the upper and lower approximation sets of with respect to , it is
unnecessary to include all the equivalence classes of them into . As shown in Table 3,
computation in cells with color is redundant. Thus, we change the initial universe set
and parameter of NSS.
Definition 7 (Paired relation soft set): Let
and
. Define the
mapping
. A pair of
is called paired relation soft set for its universe
set and parameters stemming from two relations and , respectively.
Table 4. The Tabular Representation of PRSS
\
Example
3:
For
,
example
2, let
and
and
.
.
,
and
.
The tabular representation of PRSS for the appearance and price is shown in Table 4.
Then we can construct the soft sets
and
, where mappings and are in
the formula 7. The tabular representation of soft set
and
for appearance and
price in Example 2 is shown in Table 5.
Copyright ⓒ 2016 SERSC
81
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
Table 5. The Tabular Representation of Soft Sets
on PRSS
and
appearance
Based
appearance
\
\
pric
e
1
0
0
0
pric
e
1
0
1
1
3.2. Computation of Dependency Degree Based on PRSS
According to formula 2, the computation of the dependency degree requires the lower
approximate set and the positive region. We firstly give a soft set with two parameter sets.
Then, based on that soft set and
, we present the soft set representation of the lower
approximate set and the computation of dependency degree.
A soft set with two parameter sets is shown as follows. Given an information system
=( , ) and two relations ,
,
. Let
,
. Define
mapping
. We get soft set
.
Meantime, let
,
,
,
and
.
The lower approximation set of parameter with respect to
is represented by soft
set
where
,
is as shown in Table 5. Apparently,
The lower approximation of parameter with respect to
where
(10)
.
is represented by soft set
.
Then,
Compare
(11)
.
with
, we have
=
=
(12)
where
.
The lower approximation of relation
with respect to
is represented by soft set
.
Then,
Compare
(13)
.
and
, we get,
(14)
where
.
Apparently, soft set
is the -positive region of (see formula 2 and 3).
To get the dependency degree, we also need the cardinality of
.
.
(15)
However, if we only want to get the dependency degree, it is unnecessary to calculate
first. In the following, we provide a direct way to get the dependency degree.
Based on soft set
(see table 5), the cardinality of
is
82
Copyright ⓒ 2016 SERSC
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
.
The cardinality of
(16)
is
,
The cardinality of
(17)
is
.
The degree of dependency of relation
with respect to
(18)
is
.
Given a decision system
represented by
,
(19)
, the positive region of
on
is
(20)
The dependency degree of decision
on
is represented by
.
(21)
Example 4: we present the low approximate set and the degree of dependency for
(appearance) and (price) in Example 3 based on
in Table 5.
,
,
,
.
,
,
.
The lower approximation of relation
For the calculation of cardinality,
with respect to
,
is
.
,
,
.
,
,
.
Then the dependency degree of relation
with respect to
is
.
3.3. Computation of Equivalence Classes of Indiscernibility Relations
We first introduce a new mapping as a supplement to mappings in formula 7. Then,
represent equivalence classes of the indiscernibility relation formed by two relations using
a soft set based on .
Given a PRSS
where the universal set
, parameter set
and
,
,
, we propose Definition 8.
Definition 8: Let
be the power set of ,
and
.We define
mapping
, for
The soft set based on mapping
.
Copyright ⓒ 2016 SERSC
is denoted by
.
. Let
(22)
and
,
83
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
Suppose
and
,
,
and
. Define mapping
. The equivalence classes of the indiscernibility relation formed by
are represented by
where
.
(23)
Then
.
Example 5: For (appearance) and (price) in example 3, the tabular representation
of
is shown in Table 6. Then,
.
Table 6. The Tabular Representation of
appearance
\
price
3.4. Feature Selection Algorithm Based on PRSS
Given a decision system
, the feature selection algorithm based on PRSS
is shown in Table 7.
The algorithm selects features according to the dependency degree of decision
provided by Algorithm 2 and uses the sequential forward selection [6] as the search
strategy provided by Algorithm 1. The output is a feature ranking
.
Table 7. The Feature Selection Algorithm Based on PRSS
Algorithm 1
Input
X,
Outpu
t
Algorithm 2
X is a sample set and
, and are in step 2 of
algorithm 1
, ,
is the
degree
F is a feature ranking
Begin
Step 1: Initialize
While
Step 2: Find
Step 3:
Step 4:
End
Return
End
Given the set
with
Begin
If
, then
Construct
of and based on PRSS
Calculate
Else
Construct soft set
based on
PRSS
Calculate
Construct
of
and
based on PRSS
Calculate
End
End
features selected, the th feature is determined by
.
84
dependency
(24)
Copyright ⓒ 2016 SERSC
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
Based on the ranking, we can get a feature ranking soft set
and the is defined as
, where
.
(25)
Next, we use KNN classifier to cross-validate the classification accuracy of the data
with these feature subsets. Those with the highest classification accuracy are the final
feature subsets.
Table 8. The Algorithm for the Degree of Dependency Based on NSS
Algorithm 3
Input
, ,
,
Output
and
are in step 2 in table
is the dependency degree
Begin
If
, then
Construct
Calculate
Else
Construct soft set
Calculate
Construct
Calculate
End
End
of
of
and
and
based on NSS.
according to formula 9.
based on NSS.
.
based on NSS.
.
3.5. Feature Selection Algorithm Based on NSS
Feature selection algorithm based on NSS still uses Algorithm 1 as the search strategy.
But the Algorithm 3 in Table 8 which provides the dependency degree of decision is
different from Algorithm 2.
Let
and
be the set of all equivalence classes in the partitions
and
.
Example 6: Suppose
and
.
,
.
Then
is shown in Table 9. The construct of
follows the same logic
of Table 3.
Table 9. The Tabular Representation of
price
appearance
\
price
appearance
Copyright ⓒ 2016 SERSC
85
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
4. Experimental Results
We compare the execute time of Algorithm 1 and 2 based on PRSS with that of
Algorithm 1 and 3 basd on NSS using seventeen datasets abtained from the benchmark
UCI Machine Learning Repository [22]. Those datasets are discribed in Table 10. The
field named “Number of Classes” in this table shows the aggregation of the number of
equvalence classes formed by each feature.
We run the algorithms in MATLAB R2015a on a pc with a processor Intel(R) Core(TM)
i5-2410M, a 8.0GB memory and Widows 7 operating system. Figure 1 shows datasets on
which ranges of the executive time are (0 , 0.1 ) and (0.1, 1) in seconds. Figure 2 shows
datasets on which ranges of the executive time are (1, 20) and (20, 120) in seconds.
Table 11 summrizes the comparison of feature selection algorithm based on PRSS and
NSS. It shows that PRSS improves efficiency to a certain degree on each dataset and
achieves 18.17% improvement on an average.
Figure 1. Excutive Time (Second) of Feature Selection Based on PRSS and
NSS
Table 10. Description of Datasets
No
Data sets
Number of
instances
Number of
features
Number of
Classes
1
2
3
4
Abalone
Adult
AI(Acute Inflammations )
CE(Car Evaluation)
Chess(King-Rook vs. KingPawn)
CMC(Contraceptive Method
Choice)
DER(Dermatology)
Ecoli
LC(Lung Cancer )
Lenses
Mushroom (MR)
Musk
Nursery
SFL(Solar Flare large)
SFS(Solar Flare small)
Statlog (statlog_satimage)
4177
20
120
1728
9
4
8
7
6077
10
58
25
3196
37
75
1473
10
74
366
336
32
24
8124
476
12960
1066
323
4435
34
9
55
6
23
169
9
13
13
37
135
707
154
36
119
25760
32
48
40
2752
5
6
7
8
9
10
11
12
13
14
15
16
86
Copyright ⓒ 2016 SERSC
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
17
Yeast
1484
10
1884
Figure 2. Excutive Time (Second) of Feature Selection Based on PRSS and
NSS
Figure 3. The Scalability of PRSS and NSS to the Number of Features
Table 11. Comparison of Feature Selection Algorithm of PRSS and NSS
No
Data sets
1
2
3
4
Abalone
Adult
AI(Acute Inflammations )
CE(Car Evaluation)
Chess(King-Rook vs.
5
King-Pawn)
CMC(Contraceptive
6
Method Choice)
7
DER(Dermatology)
8
Ecoli
9
LC(Lung Cancer )
10
Lenses
11
Mushroom
12
Musk
13
Nursery
14
SFL(Solar Flare large)
15
SFS(Solar Flare small)
16
Statlog(statlog_satimage)
17
Yeast
Average improvement
Copyright ⓒ 2016 SERSC
Executing time (s)
NSS
PRSS
6.645
6.620825
0.0047
0.0032
0.0128
0.008
0.0282
0.0239
Improvement
10.3424
10.0919
2.42%
0.3662
0.3511
4.10%
0.5524
0.06
0.4183
0.0052
10.42
60.4035
1.5058
0.093
0.0283
106.3707
0.9246
0.4227
0.0506
0.1632
0.0027
10.2518
56.4015
1.4757
0.078
0.0173
105.8892
0.9113
23.49%
15.68%
60.99%
47.93%
1.61%
6.63%
2.00%
16.09%
38.95%
0.45%
1.44%
18.17%
0.37%
33.50%
37.80%
15.40%
87
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
Figure 4. The Scalability of PRSS and NSS to the Number of Instances
Figure 5. The Scalability of PRSS and NSS to the Number of Equvalence
Classes
We also analyze the scalability of both algorithms on three aspects, the number of
features, the number of instances and the number of equvalence classes. Figure 3 shows
the execute time of both algorithms with regard to the number of features in each dataset.
Figure 4 shows the execute time of both algorithms with regard to the number of instances
in each dataset. We also provide the change of the execute time with regard to the
aggregation of the number of equvalence classes formed by each feature in Figure 5. In
general, three figures show the execute time increase linearly as the number of features,
instances and equvalence classes increase, respectively. Though the execute time varies at
several points, it is at an acceptabe level. Thus, those three figures demonstrate a good
scalibility of both feature selection algorithms.
5. Conclusion
Merging the soft set theory into some core conceptions of the rough set theory
facilitates the computation based on equvalence classes. We proposed a paired relation
soft set model to construct the lower approximate set, presented the lower approximate set
in the form of soft sets, and furthermore calculated the dependency degree of features. We
also provided a new mapping to calculate equivalence classes of indiscernibility relations.
Based on PRSS, the feature selection algorithm was given. Compared to the feature
selection algorithm based on NSS, the PRSS algorithm shows 18.17% improvement on an
average. Experimental results also show a good scalibility of both feature selection
algorithms. Theoretically, the paper shows a way to combine the soft set theory and the
rough set theory together to improve the feature selection algorithm. Practically, the novel
method based on PRSS can accelerate data processing procedure and is helpful in dealing
with big data.
88
Copyright ⓒ 2016 SERSC
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
In this paper, we only combined the soft set theory with the classic rough set model
which is based on equivalence classes. To deal with data with fuzzy attributes, a fuzzy
rough set model based on fuzzy similarity relations is more suitable. How to apply PRSS
into a fuzzy rough set model and facilitate the computation will be a topic of furture
reasearches.
Acknowledgment
We are very grateful to referees and editors. Their comments are very helpful in
improving the paper. We also appreciate the research team members.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
L. H. and M. H, “Feature Selection for Knowledge Discovery and Data Mining”, Kluwer, Boston,
(1998).
Z. Pawlak, “Rough Sets”, International Journal of Computer & Information Sciences, vol. 11, no. 5,
(1982), pp. 341-356.
Z. Pawlak, “Rough Sets: A Theoretical Aspect of Reasoning About Data”, Kluwer Academic Publisher,
(1991).
Z. Pawlak and A. Skowron, “Rudiments of Rough Sets”, Information Sciences, vol. 177, no. 1, (2007),
pp. 3-27.
A. Ganivada, S. S. Ray and S. K. Pal, “Fuzzy Rough Sets, and a Granular Neural Network for
Unsupervised Feature Selection”, Neural Networks, vol. 48, (2013), pp. 91-108.
Q. H. Hu, S. A. An and D. R. Yu, “Soft Fuzzy Rough Sets for Robust Feature Evaluation and Selection”,
Information Sciences, vol. 180, no. 22, (2010), pp. 4384-4400.
W. H. Shu and H. Shen, “Incremental Feature Selection Based on Rough Set in Dynamic Incomplete
Data”, Pattern Recognition, vol. 47, no. 12, (2014), pp. 3890-3906.
A. P. Zeng, T. R. Li, D. Liu, J. B. Zhang and H. M. Chen, “A Fuzzy Rough Set Approach for Incremental
Feature Selection on Hybrid Information Systems”, Fuzzy Sets and Systems, vol. 258, (2015), pp. 39-60.
S. Y. Jing, “A Hybrid Genetic Algorithm for Feature Subset Selection in Rough Set Theory”, Soft
Computing, vol. 18, no. 7, (2014), pp. 1373-1382.
B. O. Alijla, L. C. Peng, A. T. Khader and M. A. Al-Betar, “Intelligent Water Drops Algorithm for Rough
Set Feature Selection”, Intelligent Information and Database Systems (Aciids 2013), Pt Ii, vol. 7803,
(2013), pp. 356-365.
V. Rampure and A. Tiwari, “A Rough Set Based Feature Selection on Kdd Cup 99 Data Set”,
International Journal of Database Theory and Application, vol. 8, no. 1, (2015), pp. 149-156.
X. P. Gu, Y. Li and J. H. Jia, “Feature Selection for Transient Stability Assessment Based on Kernelized
Fuzzy Rough Sets and Memetic Algorithm”, International Journal of Electrical Power & Energy
Systems, vol. 64, (2015), pp. 664-670.
D. Molodtsov, “Soft Set Theory - First Results”, Computers & Mathematics with Applications, vol. 37,
no. 4-5, (1999), pp. 19-31.
G. Muhiuddin, F. Feng and Y. B. Jun, “Subalgebras of Bck/Bci-Algebras Based on Cubic Soft Sets”, The
Scientific World Journal, vol. 2014, (2014), pp. 9.
W. Xu, Z. Xiao, X. Dang, D. L. Yang and X. L. Yang, “Financial Ratio Selection for Business Failure
Prediction Using Soft Set Theory”, Knowledge-Based Systems, vol. 63, (2014), pp. 59-67.
Z. Xiao, W. J. Chen, and L. L. Li, “A Method Based on Interval-Valued Fuzzy Soft Set for MultiAttribute Group Decision-Making Problems under Uncertain Environment”, Knowledge and
Information Systems, vol. 34, no. 3, (2013), pp. 653-669.
Y. Zou and Z. Xiao, “Data Analysis Approaches of Soft Sets under Incomplete Information”,
Knowledge-Based Systems, vol. 21, no. 8, (2008), pp. 941-945.
T. Herawan and M. M. Deris, “A Direct Proof of Every Rough Set Is a Soft Set”, in 2009 3rd Asia
International Conference on Modelling and Simulation, Bandung, Bali, Indonesia, (2009), pp. 119-124.
H. W. Qin, X. Q. Ma, J. M. Zain, and T. Herawan, “A Novel Soft Set Approach in Selecting Clustering
Attribute”, Knowledge-Based Systems, vol. 36, (2012), pp. 139-145.
R. Mamat, T. Herawan and M. M. Denis, “Mar: Maximum Attribute Relative of Soft Set for Clustering
Attribute Selection”, Knowledge-Based Systems, vol. 52, (2013), pp. 11-20.
P. K. Maji, R. Biswas, and A. R. Roy, “Soft Set Theory”, Computers & Mathematics with Applications,
vol. 45, no. 4-5, (2003), pp. 555-562.
Uci Machine Learning Repository. Available: http://archive.ics.uci.edu/ml/datasets.html
Copyright ⓒ 2016 SERSC
89
International Journal of Database Theory and Application
Vol.9, No.5 (2016)
Authors
Daoli Yang, He received the B.A. degree in Electronic Business
from Shandong University in 2006. Currently, he is a Ph.D. candidate
in the School of Economics and Business Administration at
Chongqing University. His research interests include supply chain
management, information intelligent analysis and soft set theory.
Zhi Xiao, He received the B.S. degree in Department of
Mathematics from Southwest Normal University in 1982. And he
received his Ph.D. degree in Technology Economy and Management
from Chongqing University in 2003. He is currently a Professor in
the School of Economics and Business Administration at Chongqing
University. His research interests include information intelligent
analysis, data mining, and soft set theory.
Wei Xu, He received the B.A. degree in Logistic Management
from Jiangxi University of Finance and Economic in 2008. He
received his Ph.D. degree in Quantitative Economics from
Chongqing University in 2014. His research interests include
intelligent decision-making, information intelligent analysis and
business Intelligence.
Xianning Wang, He is a Ph.D. candidate in the School of
Economics and Business Administration at Chongqing
University from 2012. His research interests mainly include
dynamic economics modeling and regional economy
development.
Yuchen Pan, She is a Ph.D. candidate in the School of
Economics and Business Administration at Chongqing
University. Her research interests include information intelligent
analysis and prediction, and financial information application.
90
Copyright ⓒ 2016 SERSC